Stability and bifurcation properties of the algorithms for keeping of differential equations solutions on the required level

Algorithms of control of differential equations solutions are under investigation in the article. Idealized and real modifications of the algorithms are distinguished. An equation, which can be the base equation for investigation of the idealized algorithms properties, is constructed. The difference appearing for real systems and real algorithms is for separate investigation. This difference tends to zero under tending to zero of the time step of control. If the systems of equations satisfy or almost satisfy some properties for which the algorithms are intended, then the results are similar numerically as well. One of the algorithms demonstrates high reliability. Another one is of more complex properties. Bifurcations, periodic solutions and strange attractors are possible in both algorithms in addition to stable steady states.